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Re: IEEEP1788

Am 12.05.2015 um 13:22 schrieb John Pryce:
Ulrich

On 19 Apr 2015, at 21:35, Ulrich Kulisch <ulrich.kulisch@xxxxxxx> wrote:

In contrast to the simplicity of arithmetics in IR and IF, IEEE P1788 develops interval arithmetic on the base of IEEE 754 arithmetic with all its exceptions. This is a big mistake. It unnecessarily pulls all the IEEE 754 exceptions into interval arithmetic.

This is just not the case. At the start of the set-based flavor, with notation changed to match plain-text:

10.2. Intervals

The set of mathematical intervals provided by this flavor is denoted IR. It comprises those subsets xx of the real line R that are closed and connected in the topological sense: that is, the empty set (denoted ∅ or Empty) together with all the nonempty intervals, denoted [xlo, xhi], defined by

[xlo, xhi] = { x ∈ R | xlo ≤ x ≤ xhi }, (4) where xlo = inf xx and xhi = sup xx are extended-real numbers satisfying xlo ≤ xhi, xlo < +oo and xhi > −oo.

754 arithmetic is the dominant finite-precision approximation to R arithmetic at present which is why at Level 2 we have the idea of "754-conforming". But in 12.1.1:


An implementation shall provide at least one supported bare interval type. If 754-conforming, it shall provide the inf-sup type, see 12.5.2, of at least one of the five basic formats in 3.3 of IEEE Std 754-2008.

Thus an implementation is free to ignore 754 arithmetic entirely! So, in what way does 1788 "develop interval arithmetic on the base of IEEE 754"?

John Pryce

John:

It really makes me very sad that our conversation has reduced to some kind of nit picking.

You just pick out one paragraph of my mail of 19 April 2015. So let me repeat the three before it:

Arithmetic for R as well as for subsets F of pure floating-point numbers is well defined. On this
base arithmetic for bounded and unbounded intervals of IR and IF easily and clearly can be
derived. This leads to well known formulas which can be described on a few pages.

If division by an interval which includes zero as an interior point is excluded, interval arithmetic
leads to an exception-free, closed calculus, i.e., an operation for two intervals of IR or IF
always leads to an interval of IR resp. IF again. As an add-on division by an interval that includes
zero as an interior point also can be defined in IR and IF.  It leads to two distinct unbounded real
intervals. These can be used to develop the extended interval Newton method which allows
computing enclosures of all zeros of a function in a given domain.

Operations like oo - oo, oo/oo or 0 · oo, which in IEEE 754 arithmetic are set to NaN, do not
occur in the operations of IR and IF. For proof see my book Computer Arithmetic and Validity.
The book was published before IEEE P1788 was founded. Also the introduction of -0 or +0
does not make sense in interval arithmetic. There is only one zero in R. If zero is a bound of an
interval the other bound clearly tells whether the elements of its interior are positive or negative. 

Please have a look at the penultimate paragraph in your mail above. It reads very well. But is it
reasonable? Single precision may be too short and extended precision too slow.

Interval arithmetic carries the potential to replace floating-point arithmetic by some general
computing tool where results come with highly accurate guarantees. Two things are definitely
necessary to reach this goal:

A. Fast double precision interval arithmetic and
B. An exact dot product (EDP).

You find these two requirements already in the preface of my book "Computer Arithmetic and
Validity" (page XII). A standard that just specifies naive interval arithmetic and ignores questions of
accuracy is incomplete. It is, moreover, counterproductive since it will just reconfirm old reservations
against interval arithmetic. A simple and very fast tool for obtaining high accuracy is needed.

My interest in P1788 died when B was kicked out two years ago in 2013. A unique chance we
had to pull interval arithmetic more into the center of scientific computing was wasted!

The two letters of the IFIP Working Group on Numerical Software should not have been wiped
out so easily. I attach them once more. The EDP was available on nearly all the old mechanic calculators
as a tool for abtaining high accuracy in computing. It can be traced back to the early computer
by G. W. Leibniz (1685).

Please believe me, John,  it is not easy for me writing this mail. I admire all the work you have done.

Best wishes
Ulrich


-- 
Karlsruher Institut für Technologie (KIT)
Institut für Angewandte und Numerische Mathematik
D-76128 Karlsruhe, Germany
Prof. Ulrich Kulisch
KIT Distinguished Senior Fellow

Telefon: +49 721 608-42680
Fax: +49 721 608-46679
E-Mail: ulrich.kulisch@xxxxxxx
www.kit.edu
www.math.kit.edu/ianm2/~kulisch/

KIT - Universität des Landes Baden-Württemberg 
und nationales Großforschungszentrum in der 
Helmholtz-Gesellschaft

Attachment: IFIPWG-IEEE754R.pdf
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Attachment: IFIPWG-IEEE-P1788.pdf
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